Hyperbolic type metrics and distortion of quasiconformal map pings

This Ph.D. thesis consists of four original papers. The papers cover several topics from geometric function theory, more specifically, hyperbolic type metrics, conformal invariants, and the distortion properties of quasiconformal mappings. The first paper deals mostly with the quasihyperbolic metric. The main result gives the optimal bilipschitz constant with respect to the quasihyperbolic metric for the M¨obius self-mappings of the unit ball. A quasiinvariance property, sharp in a local sense, of the quasihyperbolic metric under quasiconformal mappings is also proved. The second paper studies some distortion estimates for the class of quasiconformal self-mappings fixing the boundary values of the unit ball or convex domains. The distortion is measured by the hyperbolic metric or hyperbolic type metrics. The results provide explicit, asymptotically sharp inequalities when the maximal dilatation of quasiconformal mappings tends to 1. These explicit estimates involve special functions which have a crucial role in this study. In the third paper, we investigate the notion of the quasihyperbolic volume and find the growth estimates for the quasihyperbolic volume of balls in a domain in terms of the radius. It turns out that in the case of domains with Ahlfors regular boundaries, the rate of growth depends not merely on the radius but also on the metric structure of the boundary. The topic of the fourth paper is complete elliptic integrals and inequalities. We derive some functional inequalities and elementary estimates for these special functions. As applications, some functional inequalities and the growth of the exterior modulus of a rectangle are studied.

Hyperbolic Type Geometries in Subdomains

This Licentiate thesis deals with hyperbolic type geometries in planar subdomains. It is known that hyperbolic type distance is always greater in a subdomain than in the original domain. In this work we obtain certain lower estimates for hyperbolic type distances in subdomains in terms of hyperbolic type distances of the original domains. In particular the domains that we consider are cyclic polygons and their circumcircles, sectors and supercircles.

Distortion properties of quasiconformal maps

In this Licentiate thesis we investigate the absolute ratio δ, j, ˜j and hyperbolic ρ metrics and their relations with each other. Various growth estimates are given for quasiconformal mpas both in plane and space. Some Hölder constants were refined with respect δ, j ˜j metrics. Some new results regarding the Hölder continuity of quasiconformal and quasiregular mapping of unit ball with respect to Euclidean and hyperbolic metrics are given, which were obtained by many authors in 1980’s. Applications are given to the study of metric space, quasiconformal and quasiregular maps in the plane and as well as in the space.

Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Forecasting of wind speeds and directions with artificial neural networks

In this master’s thesis, wind speeds and directions were modeled with the aim of developing suitable models for hourly, daily, weekly and monthly forecasting. Artificial Neural Networks implemented in MATLAB software were used to perform the forecasts. Three main types of artificial neural network were built, namely: Feed forward neural networks, Jordan Elman neural networks and Cascade forward neural networks. Four sub models of each of these neural networks were also built, corresponding to the four forecast horizons, for both wind speeds and directions. A single neural network topology was used for each of the forecast horizons, regardless of the model type. All the models were then trained with real data of wind speeds and directions collected over a period of two years in the municipal region of Puumala in Finland. Only 70% of the data was used for training, validation and testing of the models, while the second last 15% of the data was presented to the trained models for verification. The model outputs were then compared to the last 15% of the original data, by measuring the mean square errors and sum square errors between them. Based on the results, the feed forward networks returned the lowest generalization errors for hourly, weekly and monthly forecasts of wind speeds; Jordan Elman networks returned the lowest errors when used for forecasting of daily wind speeds. Cascade forward networks gave the lowest errors when used for forecasting daily, weekly and monthly wind directions; Jordan Elman networks returned the lowest errors when used for hourly forecasting. The errors were relatively low during training of the models, but shot up upon simulation with new inputs. In addition, a combination of hyperbolic tangent transfer functions for both hidden and output layers returned better results compared to other combinations of transfer functions. In general, wind speeds were more predictable as compared to wind directions, opening up opportunities for further research into building better models for wind direction forecasting.

Lihasperäiselle kreatiinikinaasille spesifinen immunomääritys Duchennen lihasdystrofian seulontaa varten

Duchennen lihasdystrofia (engl. Duchenne muscular dystrophy, DMD) on lähes pelkästään pojilla ilmenevä perinnöllinen lihasrappeumatauti, joka johtaa kuolemaan noin 25 vuoden iässä. Noin yksi 3500–6000 pojasta sairastaa DMD:tä. Taudin aiheuttaa X-kromosomissa sijaitsevan dystrofiinigeenin mutaatio, jonka seurauksena toimivaa, lihaksia koossapitävää dystrofiinia ei tuotu. Kliinisissä testeissä on lupaavia hoitoja, joten DMD:n vastasyntyneiden seulonnan aloittamista harkitaan. DMD:n seulonnassa analyyttina olisi mahdollista käyttää lihasperäistä kreatiinikinaasia (engl. muscle-type creatine kinase tai creatine kinase MM isoform, CK-MM), jota päätyy vereen lihassolujen vaurioituessa. DMD:tä sairastavilla vastasyntyneillä CK-MM:n määrä veressä on moninkertainen terveisiin vastasyntyneisiin verrattuna lihasten rappeutumisesta johtuen. Perinteisesti kreatiinikinaasia on mitattu entsyymiaktiivisuusmäärityksillä, jotka mittaavat kaikkia kreatiinikinaasimuotoja eli myös sydänperäistä ja aivoperäistä kreatiinikinaasia (CK-MB ja CK-BB). Työn tarkoituksena oli kehittää kuivatuista veritäplistä tehtävä CK-MM:lle spesifinen kaksipuoleinen immunomääritys, joka olisi siirrettävissä PerkinElmerin automaattiselle GSP® Genetic Screening Processor -analysaattorille. Työ suoritettiin kolmessa vaiheessa. Ensimmäiseksi vertailtiin kaupallisesti saatavilla olevien CK-MM-vasta-aineiden affiniteetteja biosensorilla. Seuraavassa vaiheessa pystytettiin manuaalinen kaksipuoleinen immunomääritys käyttäen ensimmäisessä vaiheessa valittuja vasta-aineita ja optimoitiin immunomäärityksen parametreja. Lopuksi immunomääritys sovitettiin GSP-laitteelle. Biosensorimittausten ja manuaalisten immunomääritysten tulosten perusteella valittiin kaksi potentiaalista leimavasta-ainetta ja yksi sitojavasta-aineeksi sopiva vasta-aine. Niitä käytettäessä määritys on melko spesifinen CK-MM:lle, sillä CK-BB ei tuottanut lainkaan signaalia ja CK-MB:n ristireaktiivisuus oli noin 7 %. GSP-laitteella mitattaessa DMD:tä sairastavien (n = 10) CK-MM-pitoisuuksien mediaani (vaihteluväli) oli 7590 ng/ml (1490–13400 ng/ml) ja terveiden vastasyntyneiden (n = 8) 165 ng/ml (108–263 ng/ml). Määrityksen dynaamista mittausaluetta ei vielä selvitetty, mutta alustavien mittausten perusteella se kattaa terveiden vastasyntyneiden pitoisuudet ja sairaiden pitoisuudet ainakin 8770 ng/ml asti, mikä mahdollistaa sairaiden erottumisen. Työssä kehitetty määritys vaikuttaa siis sopivalta DMD:n seulontaan vastasyntyneiltä.